The Axiom of Choice (AC) was formulated about a century ago,
and it was controversial for a few of decades
after that; it might be considered the last
great controversy of mathematics. It is now a basic
assumption used in many parts of mathematics. In fact,
assuming AC is equivalent to assuming
any of these principles (and many others):

Given any two sets, one set has cardinality less than or equal
to that of the other set -- i.e., one set is in one-to-one
correspondence with some subset of the other. (Historical
remark: It was questions like this that led to
Zermelo's
formulation of AC.)

Any vector space over a field F has a basis --
i.e., a maximal linearly independent subset -- over that field.
(Remark: If we only consider the case where F is the
real line, we obtain a slightly weaker statement; it is not
yet known whether this statement is also equivalent to AC.)

Any product of compact topological spaces is compact. (This is
now known as Tychonoff's Theorem, though Tychonoff himself
only had in mind a much more specialized result that is not
equivalent to the Axiom of Choice.)

AC has many forms; here
is one of the simplest:

Axiom of Choice.
Let C be a collection of nonempty sets. Then we can choose
a member from each set in that collection. In other words, there
exists a function f defined on C with the property that, for each
set S in the collection, f(S) is a member of S.

The function f is then called a choice function.

To understand this axiom better, let's consider a few
examples.

If C is the collection of all nonempty
subsets of {1,2,3,...}, then we can define f quite
easily: just let f(S) be the smallest member of S.

If C is the collection of all intervals of real
numbers with positive, finite lengths, then we can define
f(S) to be the midpoint of the interval S.

If C is some more general collection of subsets of the
real line, we may be able to define f by using a more
complicated rule.

However, if C is the collection of all nonempty
subsets of the real line, it is not clear how to find
a suitable function f. In fact, no one has ever
found a suitable function f for this collection C, and
there are convincing model-theortic
arguments that no one ever will.
(Of course, to prove this requires a precise
definition of "find," etc.)

The controversy was over how to interpret the words
"choose" and "exists" in the axiom:

If we follow the constructivists, and "exist"
means "find," then the axiom is
false, since we cannot find a choice function
for the nonempty subsets of the reals.

However, most mathematicians give "exists" a much weaker
meaning, and they consider the Axiom to be true:
To define f(S), just
arbitrarily "pick any member" of S.

In effect, when we accept the Axiom of Choice, this
means we are agreeing to the convention that we shall permit
ourselves to use a hypothetical choice function f in proofs,
as though it "exists" in some sense, even in
cases where we cannot give
an explicit example of it or an explicit algorithm for it.
(For an introduction to constructivism, you might
take a look at my
paper on that subject. The term has rather
different, slightly related meanings in advanced
mathematics and in mathematics education; I am referring
to the former meaning here.)

To assert that a mathematical object "exists," even when you cannot give an example of it, is a little bit like this: Suppose that one day you go to a football game by yourself. There are thousands of other people in the stadium, but you don't know the names of any of them. (And let's suppose you're shy, so you're not about to ask anyone their name.) Then you know those people have names, but you cannot give any of those names. (Admittedly, this is only a metaphor, and not a perfect one; don't make too much of it.)

The "existence" of f -- or of any mathematical
object, even the number "3" -- is purely formal. It
does not have the same kind of solidity as your table and your
chair; it merely exists in the mental universe of mathematics.
Many different mathematical universes are possible. When we
accept or reject the Axiom of Choice, we are specifying
something about which mental universe we're choosing to
work in. Both possibilities
are feasible -- i.e., neither accepting nor
rejecting AC yields a contradiction; that fact follows from
models devised by
Gödel
and Cohen.
However,
most "ordinary" mathematicians -- i.e., most mathematicians
who are not logicians or set theorists --
accept the Axiom of Choice chiefly
because their work is simpler with the Axiom of Choice
than without it.

Bertrand
Russell was more famous for his work in philosophy and political
activism, but he was also an accomplished mathematician.
His book Introduction to Mathematical Philosophy includes
some discussion of AC. Here is my paraphrasing of part of what he said:

To choose one sock from each of
infinitely many pairs of socks requires the Axiom of Choice,
but for shoes the Axiom is not needed.

The idea is that
the two socks in a pair are identical in appearance, and
so we must make an arbitrary choice if we wish to choose
one of them. For shoes, we can use an explicit algorithm --
e.g., "always choose the left shoe." Why does Russell's statement
mention infinitely many pairs? Well, if we only have
finitely many pairs of socks, then AC is not needed --
we can choose one member of each pair using the
definition of "nonempty," and we can repeat an operation
finitely many times using the rules of formal logic
(not discussed here).

Jerry Bona once said,

The Axiom of Choice is obviously true; the Well Ordering
Principle
is obviously false; and who can tell about Zorn's Lemma?

This is a joke. In the setting of ordinary
set theory, all three of those principles are mathematically
equivalent -- i.e., if we assume any one of
those principles, we can use it to prove the other two.
However, human intuition does not
always follow what is mathematically correct.
The Axiom of Choice agrees with the intuition of most
mathematicians; the Well Ordering Principle is contrary
to the intuition of most mathematicians; and Zorn's Lemma
is so complicated that most mathematicians are not able
to form any intuitive opinion about it.

For another indication of the controversy that initially surrounded the
Axiom of Choice, consider this anecdote (recounted by
Jan Mycielski in Notices of the AMS
vol. 53 no. 2 page 209). Tarski, one of the early great researchers in set theory and
logic, proved that AC is equivalent to the statement that any infinite set X has the
same cardinality as the Cartesian product X x X. He submitted his article to
Comptes Rendus Acad. Sci. Paris, where it was refereed by two very famous mathematicians,
Fréchet and Lebesgue. Both wrote letters rejecting the article. Fréchet wrote that
an implication between two well known truths is not a new result. And
Lebesgue wrote that an
implication between two false statements is of no
interest. Tarski said that he never again submitted a paper to the Comptes Rendus.

AC permits arbitrary choices from an arbitrary
collection of nonempty
sets.
Some mathematicians
have investigated some weakened forms of AC, such as

CC (Countable Choice), which permits arbitrary
choices from a sequence
of nonempty sets.

DC (Dependent Choice), which permits
the more general process of selecting arbitrarily
from a sequence of nonempty sets where only
the first set is specified in advance; each
subsequent set of options may depend somehow on the
previous choices.
This is precisely what
is needed for some choice processes in topology and analysis --
e.g., for the proof of the Baire Category Theorem.

The full strength of the Axiom of Choice does
not seem to be needed for applied mathematics.
Some weaker principle such as CC or DC
generally would suffice. To see this, consider that
any application is based on measurements, but humans
can only make finitely many measurements.
We can extrapolate and take limits, but usually
those limits are sequential, so even in theory we
cannot make use of more than countably many
measurements. The resulting spaces are separable.
Even if we use a nonseparable space such as
L∞, this
may be merely to simplify our notation; the
relevant action may all be happening in some
separable subspace, which we could identify
with just a bit more effort. (Thus, in some
sense, nonseparable spaces exist only
in the imagination of mathematicians.) If we restrict
our attention to separable spaces, then much
of conventional analysis still works with
AC replaced by CC or DC. However, the
resulting exposition is then more complicated,
and so this route is only followed by a few
mathematicians who have strong philosophical
leanings against AC.

A few pure mathematicians and many applied mathematicians
(including, e.g., some mathematical physicists) are
uncomfortable with the Axiom of Choice. Although
AC simplifies some parts of mathematics, it also
yields some results that are unrelated to, or perhaps even
contrary to, everyday "ordinary" experience; it implies the
existence of some rather bizarre, counterintuitive
objects. Perhaps the most bizarre is the
Banach-Tarski Paradox:
It is possible to take the 3-dimensional
closed unit ball,

B = {(x,y,z)
∈ R3
: x2 + y2 + z2< 1}

and partition it into finitely many pieces, and move those
pieces in rigid motions (i.e., rotations and translations,
with pieces permitted to move through one another) and
reassemble them to form two copies of B.

At first glance, the Banach-Tarski result
seems to contradict some of our intuition about physics
-- e.g., the Law of Conservation of Mass, from classical
Newtonian physics. If we assume that the ball has
a uniform density, then the Banach-Tarski Paradox
seems to say
that we can disassemble a one-kilogram ball
into pieces and rearrange them to get two
one-kilogram balls. But actually, the
contradiction can be explained away:
Only a
set with a defined volume can have a defined mass.
A "volume" can be defined for many subsets
of R3 ---
spheres, cubes, cones, icosahedrons, etc. --- and in fact
a "volume" can be defined for nearly any subset of
R3 that we can think of.
This leads beginners to expect that the notion
of "volume" is applicable to
every subset of R3.
But it's not. In particular, the pieces in the
Banach-Tarski decomposition are sets whose
volumes cannot be defined.

More precisely, Lebesgue measure
is defined on some subsets of R3, but
it cannot be extended to all subsets of
R3 in a fashion that preserves two of
its most important properties: the measure of the union
of two disjoint sets is the sum of their measures, and
measure is unchanged under translation and rotation.
The pieces in the Banach-Tarski decomposition
are not Lebesgue measurable. Thus, the Banach-Tarski
Paradox
gives as a corollary the fact that there exist
sets that are not Lebesgue measurable.
That corollary also has a much shorter proof (not
involving the Banach-Tarski Paradox)
which can be found
in every introductory textbook on
measure theory, but it too uses the
Axiom of Choice.

Here is a brief sketch of that shorter proof:
Work in "the real numbers modulo 1" -- that is, the number system that you get if
you cut the interval [0,1) out of the real line and loop it around into a circle, so that
0 and 1 are the same number. (Like the way that 0 and 12 are the same on a circular clock.)
In that number system, multiplication and division don't really work very well any more, but
addition and subtraction still work fine, and so does Lebesgue measure. Let's call that number system
T; its entire measure is 1. Now, the
Axiom of Choice is used to "construct" a rather peculiar subset of T -- let us call it C -- with the property that the sets
C+r = {x+r : x in C} are all disjoint from each other, for different values of the
rational number r. The union of these sets is all of T. Now, if C were measurable, then so would each C+r, and they would
all have the same measure, and their measures would add up to the measure of T -- that is, they would add up to 1. But how many of these C+r are there? There are a countable infinity of them. If the measure of C were zero, their sum would be zero.
If the measure of C were positive, their sum would be infinite. You can't get 1, either way.

Personally, I am not surprised to find the Axiom of Choice
coming into play in a subject that is so inherently
complicated as unmeasurable sets.
I am much more surprised to find AC coming into play
in this simpler and more concrete example:
I want to classify all subsets of {1, 2, 3, 4, 5, . . .} as "small" or
not "small,"
defining the word "small" in such a way that

any set with zero or one members is "small";

any union of two "small" sets is "small"; and

a set is"small" if and only if its complement isn't "small."

Now, without much difficulty I can give examples satisfying any two of those
three rules:

Define "small" to mean "finite." This satisfies rules a and b. But it
does not satisfy rule c, since the even numbers and the odd numbers are complements
of each other, and neither of those sets is finite.

Say that a set is "small" if the number 1 is not a member of that set.
This definition satisfies rules b and c, but it classifies the set {1} as "not small," thus
failing rule a.

Say that a set is "small" if it contains at most one of the three numbers 1, 2, 3.
That satisfies rules a and c. But it classifies the sets {1} and {2} as small and the
set {1,2} as not small, thus failing rule b.

Does there exist a classification scheme satisfying all three rules?
It turns out that such a classification scheme exists, but an
example of such a classification scheme does not exist
(which makes it a bit hard to visualize!).
And by that I do not mean just that we haven't found an example yet. I mean
that the proofs of existence are inherently nonconstructive -- i.e., they
cannot be replaced by constructive proofs -- so no
examples can ever be given. But the proof of that fact is very deep,
and it raises interesting philosophical questions: In what sense does
that classification scheme "exist"?
(My own attitude is that I'm not really working with the classification
schemes themselves;
I'm just working
with sentences about hypothetical classification
schemes.)

Technical details for experts:
To prove the existence of such a classification scheme, just
call "large" the members of some nonprincipal ultrafilter
on the positive integers, and call their complements "small."
Note that, with this scheme, any superset of a "large" set
is also "large."
The converse is slightly more complicated: If you have
a "small/large" classification, the "large" sets do not necessarily
necessarily form a nonprincipal ultrafilter, but the supersets
of "large" sets do.
An introduction to nonprincipal ultrafilters can be
found in my book and in many other places in the
literature.
The existence of nonprincipal ultrafilters follows easily
from Zorn's Lemma, by arguments that will be obvious
once you've digested all definitions involved
(admittedly not a small task).
But showing that the existence proof
is inherently nonconstructive is much harder,
and requires some definitions that I've made up.
By an "example" I mean anything whose existence
can be proved using just ZF+DC --- that is, I'll
allow Dependent Choice but no higher relatives
of AC.
Let BP be the statement that "every subset
of the reals has the Baire property."
The existence of a nonpricipal ultrafilter
on the integers implies not-BP (by fairly
straightforward
functional analysis and topology).
But in 1984 Shelah proved that
the consistency of ZF implies the
consistency of ZF+DC+BP. Therefore,
if ordinary set theory is free of contradictions,
then ZF+DC cannot be used
to prove not-BP.
I say "if" because we don't know that for sure,
and Gödel's Incompleteness Theorem
assures us that we never will know the
consistency of ZF for sure. However,
I would say that ZF is empirically consistent:
In a century of study, mathematicians have
not yet found any contradictions in ZF, despite
the incentive that any mathematician finding such
an important proof would instantly be promoted to full
professor at any university in the world.
My example with positive integers might
appear to be simpler than the
Banach-Tarski Paradox, but it
does not really get us completely away from measure
theory. A nonprincipal ultrafilter can be reformulated
as a two-valued probability measure
that is finitely additive but not countably additive.

In the preceding paragraphs I have attempted to
introduce the Axiom of Choice in the language of
informal set theory (also known as "naive set theory"), in
which one assumes that sets are
"collections of objects," with the meanings of the
words "collection" and "object"
based on our everyday nonmathematical experience.
However, this web page is merely an introduction to the subject,
and gives no indication of the proofs. A
mathematically precise study of AC would require formal set
theory (also known as "axiomatic set theory"); that
is the language spoken by the real experts in this subject.
In formal set theory, we put aside any nonmathematical,
notions of "set". Instead of working with imagined sets,
we work with sentences about some objects that we
call "sets." We begin with a list of axioms,
i.e., properties that these abstract objects are assumed
to satisfy. Generally we assume axioms that seem intuitively
reasonable, but we assume as few axioms as possible --
we try to choose axioms that enable us to prove the
other properties that we want.
We assume nothing else beyond what is
contained in those axioms; we may not use a property
of sets just because it is "familiar" or "obvious."
Then we study the nature of the proofs, to determine
what kinds of things can or cannot be proved.
The axiomatic approach is much drier, and less appealing
to all but a few specialists who are interested in it;
but its conclusions are much more reliable than
any mere, informal discussion. In this web page I have
attempted to summarize, in informal
language, some of the conclusions that are reached
by that formal theory.

Links Collection for AC

Please write to me
if you have suggestions for additions or alterations to this
web page. However, I will warn you that I am NOT a leading
authority on the Axiom of Choice; I am not knowledgeable about much of
the advanced research on the subject. I have posted
this web page chiefly because
(i) I like the Axiom of Choice;
(ii) I think I have a good
understanding of the elementary aspects of the subject;
and (iii) I like posting web
pages.

Introductory / elementary

For more extensive information about the Banach-Tarski
Paradox, see
Stan Wagon's
book.

Handbook of
Analysis and its Foundations, by Eric Schechter.
The website is an advertisement,
but it does include a few interesting excerpts from the book
-- e.g., a
list of 27 forms of the Axiom of Choice
and a few dozen weak forms of Choice, as well as
a chart showing how some of the
weak forms are related. (The book is
intended for beginning graduate students;
only a small portion of the book
is actually concerned with Choice.)

Zermelo's axiom of choice : its origins, development, and
influence, by Gregory Moore. A fascinating history of AC. Originally published by Springer, now available as an inexpensive reprint from Dover.
Here is a web page giving the
table of
contents of that book.

Constructivism
is Difficult -- a brief introduction to constructivism.
Constructivism and AC are two different but overlapping topics.
AC is a nonconstructive axiom; that's what made it so
controversial.

One of AC's most important applications
in analysis is the Hahn-Banach Theorem. It may be viewed as
a weak form of Choice. Here is an on-line survey article,
The
Hahn-Banach Theorem: The Life and Times, by
Lawrence Narici and Edward Beckenstein.

Axiome
du Choix, by
David
Madore. This page
includes a list of several equivalents and weaker consequences of
AC, and a list of some of the implications among them.
If your French is weak, you might try an automatic translation
service, such as
AV's translator.
However, such translation programs don't know mathematics, and so
some of the results may be a bit odd. For instance,
what we call "well ordering" is what the French call
"bon ordre," but the AV translator turns that into
"good command."

An
introduction to the generalized Riemann integral, by me (Eric Schechter).
This integral -- actually more general than the Lebesgue integral,
but much simpler to define --
is a good topic to include in the "introduction to real
analysis" course that many universities teach to advanced
undergraduates majoring in math.
Near the end of the web page,
the Axiom of Choice arises naturally in a discussion
about the measurability of sets.

Especially noteworthy books and/or researchers

Consequences
of the Axiom of Choice is a book by Paul Howard and Jean E. Rubin
that was published by the American Mathematical
Society in 1998. It is a vast survey of Choice and its weaker
relatives. It is a reference book, not intended
for beginners. The authors
are continuing their research project, which now goes a bit beyond
the book. Their web page contains a list of the errata and addenda
to the book, and a form for downloading copies of the project's
main tables. You may also want to look at some related
papers.

Home page of Thomas Jech.
Jech is the author of the book titled The Axiom of Choice, which
is not recent but is still excellent. He has worked in
set theory, logic, and other areas since then. Some of his
papers are available online.

Saharon Shelah's papers.
Shelah is one of the leading logicians of our century; he has made
great contributions to the theory of forcing. My favorite among his
results is the fact that Con(ZF) implies Con(ZF + DC + BP); this
result was shown by J.D.M.Wright to be important to functional
analysis. (It's explained further in my book.) Among Shelah's
subpages is a
list of his coauthors,
many of whom have web pages of their own.

Andreas
Blass's home page. It is a very old result that the Axiom
of Choice implies the existence of bases for vector spaces;
Blass can be credited with proving the converse. Blass also
did some of the early work on proving the unconstructability
of nonprincipal ultrafilters. (Those are results of his
that I've understood; he has probably done some other much
more important things that I don't understand.)

Edward Nelson's
home page. Nelson is the father of Internal Set Theory (IST), a
variant of Nonstandard Analysis. IST has acquired a large
following; some analysts are of the opinion that IST is the most
intuitive approach to limits. (Personally,
I suspect that most of those analysts were first trained
as logicians, but that may be a reflection of my own ignorance.)
Some of Nelson's writings are available online.

Home page of John L. Bell,
another leading researcher. Bell's page includes many interesting links
and downloadable papers. Among those papers,
Axiom of Choice and Zorn.pdf
may be of particular interest to beginning and intermediate readers
about AC. It includes (among other things) a chronology of choice principles,
a chronology of maximal principles, and a list of many choice principles.

New
Foundations
home page, by Randall Holmes. NF is a refinement of
Russell's theory of types, introduced by Quine in 1937. Thus, it is an
alternate form of set theory or higher-order logic, a little different
from conventional set theory, but still capable of doing approximately
the same things. Of course, it differs from the usual set theory in a
few respects; an obvious difference is that there is a universal set
in NF. A surprising difference is that the Axiom of Choice is false
in NF; this was established in a 1953 paper by E.P.Specker. If
Holmes's website interests you, you might continue with T.E.Forster's
1995 book. (For bibliographic details see Holmes's website.)

Issues
in commonsense set theory, an online article by Mujdat Pakkan and Varol
Akman. From the abstract: "In this survey, we briefly review classical
set theory from an AI perspective, and then consider alternative set theories."
Includes an overview of ZF set theory, which makes it relevant to this
home page.

Formal logic and / or automatic theorem-proving

Metamath
Solitaire, an elementary game implemented as a Java applet
that lets you prove simple theorems in logic and set theory.
Includes introductory explanation.

Isabelle
is a generic theorem prover which can support a wide variety of logics;
it is available for free and will run on most Unix systems.
You may be interested in some of
Larry
Paulson's papers on Isabelle.
Especially, you may be interested in "Mechanising
Set Theory: Cardinal Arithmetic and the Axiom of Choice", coauthored by
Krzysztof Grabczewski. This paper mechanizes
the proof of numerous equivalents of the Axiom of Choice, covering
most of Chapter 1 of Kunen's Set Theory and most of
Chapters 1 and 2 of Rubin and Rubin's Equivalents of the Axiom
of Choice.

Miscellaneous

The Axiom of Choice was used for
a tongue-in-cheek "proof" of the existence of God, by
Robert K. Meyer in
"God exists!", Nous21 (1987), 345-361.
The basic idea is to put a suitable
partial ordering on the universe, and
then use Zorn's Lemma to prove the existence
of a maximal element, which is therefore God. A
web page
reprinting the proof has been made available by
Alexander Pruss.

A musical band named Axiom of Choice.
Their music is a fusion of Persian Traditional with modern. Okay, it's
not math, but I couldn't resist posting this here.
Iranian born guitarist Ramin Torkian and singer Mamek Khadem
were both trained as mathematicians, and
that's where the group gets its name. Their first album's
liner notes say
"There is an exciting and profound artistic value in the
mathematical principle, Axiom of Choice. The mathematician
has the right to choose elements without explanation. In a world
where everything must be explained, these choices are voluntary
and do not need explanation."